Outline of The Theory
We now expound Krause's (1992) axiomatic theory, the first quasi-set theory; other formulations and improvements have since appeared. For an updated paper on the subject, see French and Krause (2010). Krause builds on the set theory ZFU, consisting of Zermelo-Fraenkel set theory with an ontology extended to include two kinds of urelements:
- m-atoms, whose intended interpretation is elementary quantum particles;
- M-atoms, macroscopic objects to which classical logic is assumed to apply.
Quasi-sets (q-sets) are collections resulting from applying axioms, very similar to those for ZFU, to a basic domain composed of m-atoms, M-atoms, and aggregates of these. The axioms of include equivalents of extensionality, but in a weaker form, termed "weak extensionality axiom"; axioms asserting the existence of the empty set, unordered pair, union set, and power set; Separation; the image of a q-set under a q-function is also a q-set; q-set equivalents of Infinity, Regularity, and Choice. Q-set theories based on other set-theoretical frameworks are, of course, possible.
has a primitive concept of quasi-cardinal, governed by eight additional axioms, intuitively standing for the quantity of objects in a collection. The quasi-cardinal of a quasi-set is not defined in the usual sense (by means of ordinals) because the m-atoms are assumed (absolutely) indistinguishable. Furthermore, it is possible to define a translation from the language of ZFU into the language of in such a way so that there is a 'copy' of ZFU in . In this copy, all the usual mathematical concepts can be defined, and the 'sets' (in reality, the '-sets') turn out to be those q-sets whose transitive closure contains no m-atoms.
In there may exist q-sets, called "pure" q-sets, whose elements are all m-atoms, and the axiomatics of provides the grounds for saying that nothing in distinguishes the elements of a pure q-set from one another, for certain pure q-sets. Within the theory, the idea that there is more than one entity in x is expressed by an axiom which states that the quasi-cardinal of the power quasi-set of x has quasi-cardinal 2qc(x), where qc(x) is the quasi-cardinal of x (which is a cardinal obtained in the 'copy' of ZFU just mentioned).
What exactly does this mean? Consider the level 2p of a sodium atom, in which there are six indiscernible electrons. Even so, physicists reason as if there are in fact six entities in that level, and not only one. In this way, by saying that the quasi-cardinal of the power quasi-set of x is 2qc(x) (suppose that qc(x) = 6 to follow the example), we are not excluding the hypothesis that there can exist six subquasi-sets of x which are 'singletons', although we cannot distinguish among them. Whether there are or not six elements in x is something which cannot be ascribed by the theory (although the notion is compatible with the theory). If the theory could answer this question, the elements of x would be individualized and hence counted, contradicting the basic assumption that they cannot be distinguished.
In other words, we may consistently (within the axiomatics of ) reason as if there are six entities in x, but x must be regarded as a collection whose elements cannot be discerned as individuals. Using quasi-set theory, we can express some facts of quantum physics without introducing symmetry conditions (Krause et al. 1999, 2005). As is well known, in order to express indistinguishability, the particles are deemed to be individuals, say by attaching them to coordinates or to adequate functions/vectors like |ψ>. Thus, given two quantum systems labeled |ψ1> and |ψ2> at the outset, we need to consider a function like |ψ12> = |ψ1>|ψ2> ± |ψ2>|ψ1> (except for certain constants), which keep the quanta indistinguishable by permutations; the probability density of the joint system independs on which is quanta #1 and which is quanta #2. (Note that precision requires that we talk of "two" quanta without distinguishing them, which is impossible in conventional set theories.) In, we can dispense with this "identification" of the quanta; for details, see Krause et al. (1999, 2005) and French and Krause (2006).
Quasi-set theory is a way to operationalize Heinz Post's (1963) claim that quanta should be deemed indistinguishable "right from the start."
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